Softening of First-Order Phase Transition on Quenc

arXiv:hep-lat/9607033v116Jul1996MainzpreprintKOMA-96-21SofteningofFirst-OrderPhaseTransitiononQuenchedRandomGravityGraphsCliveF.Baillie1,WolfhardJanke2andDesmondA.Johnston31DepartmentofComputerScience,UniversityofColoradoBoulder,CO80309,USA2Institutf¨urPhysik,JohannesGutenberg-Universit¨atMainzStaudingerWeg7,55099Mainz,Germany3DepartmentofMathematics,Heriot-WattUniversityEdinburgh,EH144AS,ScotlandAbstractWeperformextensiveMonteCarlosimulationsofthe10-statePottsmodelonquenchedtwo-dimensionalΦ3gravitygraphstostudytheeffectofquenchedcoordinationnumberrandomnessonthenatureofthephasetransition,whichisstronglyfirstorderonregularlattices.Thenumericaldataprovidesstrongevidencethat,duetothequenchedrandomness,thediscontinuousfirst-orderphasetransitionofthepuremodelissoftenedtoacontinuoustransition,representingpresumablyanewuniversalityclass.Thisresultisinstrikingcontrasttoare-centMonteCarlostudyofthe8-statePottsmodelontwo-dimensionalPoissonianrandomlatticesofVoronoi/Delaunaytype,wherethephasetransitionclearlystayedoffirstorder,butisinqualitativeagreementwithresultsforquenchedbondrandomnessonregularlattices.Aprecedentforsuchsofteningwithconnectivitydisorderisknown:inthe10-statePottsmodelonannealedΦ3gravitygraphsacontinuoustransitionisalsoobserved.1IntroductionSystemssubjecttoquenchedrandomdisorderoftenshowacompletelydif-ferentbehaviorthaninthepurecase.Ifthepuresystemhasacontinuousphasetransitionitiswellknownthatquenchedrandomdisordercandrivethecriticalbehaviorintoanewuniversalityclass,orthetransitioncanevenbeeliminatedaltogether.1Inthecaseofafirst-orderphasetransitioninthepuresystemtheeffectofquenchedrandomdisordercanalsobeverydra-matic.Infact,phenomenologicalrenormalization-groupargumentssuggestthepossibilityforasofteningtoacontinuoustransition.2Theparadigmfortestingthelatterpredictionisthetwo-dimensional(2D)q-statePottsmodel.Thismodelisexactlyknown3toexhibitonregularlatticesforq≥5afirst-ordertransitionwhosestrengthincreaseswithq.InRef.4theeffectofquenchedbonddisorderwasinvestigatedforthe8-statemodel.BymeansofextensiveMonteCarlo(MC)simulationsthepredictedsofteningwasconfirmed,andafinite-sizescaling(FSS)analysisshowedthatthecriticalbehaviorofthequenchedmodelcouldbewelldescribedbytheOnsagerIsingmodeluniversalityclass.4InRefs.5,6theeffectofquenchedconnectivitydisorderwasstudiedbyputtingthe8-statemodelon2DPoissonianrandomlatticeswithtoroidaltopology,constructedaccordingtotheVoronoi/Delaunayprescription.7Herethebondsareallofequalstrength,butthedistributionofcoordinationnumbers(=3,...,∞)variesrandomlyfromlatticetolattice,givingrisetothequenchedrandomdisorder.MCsimulationscombinedwithFSSanalysesprovidedclearevidencethatforthistypeofquencheddisorderthetransitionstaysfirstorderasinthepurecase.Adifferent,stronger1,sortofcoordinationnumberrandomnessappearsin2DgravitytriangulationsortheirdualΦ3graphs.Insuchmodelsoneisinterestedinthecouplingofmatterto2Dgravity,sothedisorderisannealedratherthanquenched.MotivatedbyWexler’smeanfieldresultsforq=∞Pottsmodelscoupledto2Dgravity,8simulationsofthe10-stateand200-statePottsmodelcoupledto2Dgravity(i.e.,onanannealedensembleofΦ3graphsofsphericaltopology)gaveconvincingevidence9foracontinuoustransition,withthemeasuredcriticalexponentsforthe10-statePottsmodelbeingconsistentwiththeKPZexponentsofthe4-statePottsmodelcoupled1Weprovidesomejustificationfortheuseof“stronger”inwhatfollows.1to2Dgravity.Astheonlyquenchedconnectivitydisorderseriouslyinvestigatedtodate,2DPoissonianrandomlattices,showednosignofsofteningforfirst-ordertransitionsitisinterestingtoenquirewhetherthesalientfeatureforthesoft-eninginthe2Dgravitysimulationsistheannealednatureoftheconnectivitydisorderorwhetheritissomeintrinsicfeaturesofthegraphsthemselves.ThesimulationsalsotouchonthequestionofwhatconstitutestheuniversalityclassoftheKPZexponentsformattercoupledto2Dgravity.Recentworkhassuggestedthatthefull2DgravitycurvaturedistributionisnotrequiredondynamicaltriangulationsinordertostaywithintheKPZuniversalityclass.10,11Thecurrentworkcanalsobeviewedasaddressingthequestionofwhetherquenchingthedynamicallatticeshasanyeffectontheexponents.InthisnotewepresentresultsofaMCstudyonquenchedrandomlatticesdrawnfromtheequilibriumdistributionofpure2Dgravitytriangulations.Tobeprecisewesimulatedthe10-statePottsmodelonthedualofthesetriangulations,theso-calledΦ3graphsofsphericaltopology.Asourmainresultweobtainstrongevidencethatforthegravitytypeofrandomlat-ticesthetransitionisindeedsoftenedandseemstodefineanew“quenched”universalityclass.2ModelandsimulationWeusedthestandarddefinitionoftheq-statePottsmodel,ZPotts=X{σi}e−βE;E=−Xhijiδσiσj;σi=1,...,q,(1)whereβ=J/kBTistheinversetemperatureinnaturalunits,δistheKro-neckersymbol,andhijidenotesthenearest-neighbourbondsofrandomΦ3graphs(withouttadpolesorself-energybubbles)withN=250,500,1000,2000,3000,5000,and10000sites.Foreachlatticesizewegenerated64independentreplicausingtheTuttealgorithm,12andperformedlongsimula-tionsofthe10-statemodelnearthetransitionpointatˆβ=2.20,2.20,2,20,2.22,2.23,2.24,and2.242,respectively,usingthesingle-clusteru